Group #27142561 — subgroups

Subgroups — #27142561

Order 12416.

412 conjugacy classes of subgroups; only the 113 normal subgroups are shown.

normal subgroups only

This group has 412 conjugacy classes of subgroups — too many to draw. Only its 113 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#15690731	C₂	1	4	yes	maximal, complemented
2	C₂×#15690721	C₂	1	1	yes	maximal, characteristic, complemented
3	#20260601	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×#127×C₉₇	C₂	1	1	yes	maximal, characteristic
5	C₂²×#11715895	C₂²	1	2	yes	—
6	C₂×C₄⋊Dic₉₇	C₂²	1	2	yes	—
7	#15690731	—	2	8	no	—
8	#15690721	—	2	2	no	—
9	#15690717	—	2	2	no	—
10	C₂²×D₄×C₉₇	—	2	2	no	—
11	C₂²×D₄×C₉₇	C₂²	1	2	yes	—
12	#127×C₉₇	—	2	2	no	—
13	#127×C₉₇	C₄	1	4	yes	complemented
14	C₂×C₄²×C₉₇	C₂²	1	1	yes	characteristic
15	C₂²×Dic₉₇	—	2	2	no	—
16	C₂×#11715895	—	2	2	no	—
17	C₂×#11715895	—	2	2	no	—
18	C₂×#11715895	D₄	1	4	yes	complemented
19	C₄⋊Dic₉₇	—	2	4	no	—
20	C₂⁴×C₉₇	—	2	2	no	—
21	C₂×D₄×C₉₇	—	2	8	no	—
22	C₂×D₄×C₉₇	—	2	4	no	—
23	C₂×D₄×C₉₇	—	2	2	no	—
24	C₂×D₄×C₉₇	—	2	4	no	—
25	C₂×D₄×C₉₇	—	4	8	no	—
26	C₂×D₄×C₉₇	C₂×C₄	1	4	yes	—
27	C₂²×C₄×C₉₇	D₄	1	2	yes	—
28	C₂²×C₄×C₉₇	C₂³	1	1	yes	characteristic
29	C₄²×C₉₇	C₂×C₄	1	2	yes	—
30	C₄²×C₉₇	D₄	1	2	yes	—
31	C₂³×C₉₇	—	2	4	no	—
32	C₂³×C₉₇	—	4	2	no	—
33	C₂³×C₉₇	—	4	2	no	—
34	C₂³×C₉₇	—	4	8	no	—
35	C₂³×C₉₇	C₂×D₄	1	1	yes	characteristic
36	D₄×C₉₇	—	2	8	no	—
37	D₄×C₉₇	—	4	16	no	—
38	D₄×C₉₇	—	4	4	no	—
39	C₂×C₄×C₉₇	—	2	4	no	—
40	C₂×C₄×C₉₇	C₂²⋊C₄	1	4	yes	—
41	C₂×C₄×C₉₇	C₂²×C₄	1	1	yes	characteristic
42	C₂×C₄×C₉₇	C₂×D₄	1	2	yes	—
43	C₂×C₄×C₉₇	C₂×D₄	1	1	yes	characteristic
44	C₂×C₄×C₉₇	#41	1	2	yes	—
45	C₂×Dic₉₇	—	4	2	no	—
46	C₂×Dic₉₇	—	4	4	no	—
47	#11715895	—	2	4	no	—
48	C₂²×C₉₇	—	4	4	no	—
49	C₂²×C₉₇	—	4	8	no	—
50	C₂²×C₉₇	—	8	4	no	—
51	C₂²×C₉₇	—	8	4	no	—
52	C₂²×C₉₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
53	C₂²×C₉₇	#124	1	1	yes	characteristic
54	C₂²×C₉₇	#127	1	1	yes	characteristic
55	C₂²×C₉₇	AGL(1,ℤ/8)	1	4	yes	—
56	C₄×C₉₇	—	2	4	no	—
57	C₄×C₉₇	C₄×D₄	1	2	yes	—
58	C₄×C₉₇	#121	1	2	yes	—
59	Dic₉₇	—	8	2	no	—
60	C₂×C₉₇	—	8	4	no	—
61	C₂×C₉₇	#390	1	1	yes	characteristic
62	C₂×C₉₇	#418	1	2	yes	—
63	C₂×C₉₇	#490	1	2	yes	—
64	C₂×C₉₇	#496	1	2	yes	—
65	#1961	—	97	1	no	maximal, Sylow
66	C₉₇	#1961	1	1	yes	characteristic, complemented, Sylow
67	C₂×#127	D₉₇	1	1	yes	characteristic
68	C₂×#102	—	97	4	no	—
69	C₂×#97	—	97	1	no	—
70	#382	—	97	1	no	—
71	C₂²×D₄	—	2	2	no	—
72	C₂²×D₄	C₂×D₉₇	1	2	yes	—
73	C₂²×C₈	—	97	2	no	—
74	#127	—	2	2	no	—
75	#127	Dic₉₇	1	4	yes	complemented
76	C₂×C₄⋊C₄	—	97	2	no	—
77	C₂×C₄²	C₂×D₉₇	1	1	yes	characteristic
78	#102	—	194	8	no	—
79	#97	—	194	2	no	—
80	#95	—	194	2	no	—
81	C₂⁴	—	2	2	no	—
82	C₂×D₄	—	2	8	no	—
83	C₂×D₄	—	2	4	no	—
84	C₂×D₄	—	2	2	no	—
85	C₂×D₄	—	2	4	no	—
86	C₂×D₄	—	4	8	no	—
87	C₂×D₄	C₂×Dic₉₇	1	4	yes	—
88	C₂²×C₄	—	194	2	no	—
89	C₂²×C₄	C₂²⋊D₉₇	1	2	yes	—
90	C₂²×C₄	C₂²×D₉₇	1	1	yes	characteristic
91	C₂×C₈	—	97	4	no	—
92	C₂×C₈	—	194	2	no	—
93	C₂×C₈	—	194	2	no	—
94	C₄⋊C₄	—	194	4	no	—
95	C₄²	C₂×Dic₉₇	1	2	yes	—
96	C₄²	C₂²⋊D₉₇	1	2	yes	—
97	C₂³	—	2	4	no	—
98	C₂³	—	4	2	no	—
99	C₂³	—	4	2	no	—
100	C₂³	—	4	8	no	—
101	C₂³	C₂×C₂²⋊D₉₇	1	1	yes	characteristic
102	D₄	—	2	8	no	—
103	D₄	—	4	16	no	—
104	D₄	—	4	4	no	—
105	C₂×C₄	—	2	4	no	—
106	C₂×C₄	—	388	2	no	—
107	C₂×C₄	—	388	4	no	—
108	C₂×C₄	C₂²⋊Dic₉₇	1	4	yes	—
109	C₂×C₄	C₂²×Dic₉₇	1	1	yes	characteristic
110	C₂×C₄	D₄×D₉₇	1	2	yes	—
111	C₂×C₄	C₂×C₂²⋊D₉₇	1	1	yes	characteristic
112	C₂×C₄	#13095510	1	2	yes	—
113	C₈	—	194	4	no	—
114	C₂²	—	4	4	no	—
115	C₂²	—	4	8	no	—
116	C₂²	—	8	4	no	—
117	C₂²	—	8	4	no	—
118	C₂²	C₂×C₂²⋊Dic₉₇	1	1	yes	characteristic
119	C₂²	#15690797	1	1	yes	characteristic
120	C₂²	#15690806	1	1	yes	characteristic
121	C₂²	#15690836	1	4	yes	—
122	C₄	—	2	4	no	—
123	C₄	—	776	2	no	—
124	C₄	D₄×Dic₉₇	1	2	yes	—
125	C₄	#15690781	1	2	yes	—
126	C₂	—	8	4	no	—
127	C₂	#20260638	1	1	yes	characteristic
128	C₂	#20260740	1	2	yes	—
129	C₂	#20261099	1	2	yes	—
130	C₂	#20261122	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#15690731

C₂

yes

maximal, complemented

C₂×#15690721

C₂

yes

maximal, characteristic, complemented

#20260601

C₂

yes

maximal, characteristic, complemented

C₂×#127×C₉₇

C₂

yes

maximal, characteristic

C₂²×#11715895

C₂²

yes

—

C₂×C₄⋊Dic₉₇

C₂²

yes

—

#15690731

—

#15690721

—

#15690717

—

C₂²×D₄×C₉₇

—

C₂²×D₄×C₉₇

C₂²

yes

—

#127×C₉₇

—

#127×C₉₇

C₄

yes

complemented

C₂×C₄²×C₉₇

C₂²

yes

characteristic

C₂²×Dic₉₇

—

C₂×#11715895

—

C₂×#11715895

—

C₂×#11715895

D₄

yes

complemented

C₄⋊Dic₉₇

—

C₂⁴×C₉₇

—

C₂×D₄×C₉₇

—

C₂×D₄×C₉₇

—

C₂×D₄×C₉₇

—

C₂×D₄×C₉₇

—

C₂×D₄×C₉₇

—

C₂×D₄×C₉₇

C₂×C₄

yes

—

C₂²×C₄×C₉₇

D₄

yes

—

C₂²×C₄×C₉₇

C₂³

yes

characteristic

C₄²×C₉₇

C₂×C₄

yes

—

C₄²×C₉₇

D₄

yes

—

C₂³×C₉₇

—

C₂³×C₉₇

—

C₂³×C₉₇

—

C₂³×C₉₇

—

C₂³×C₉₇

C₂×D₄

yes

characteristic

D₄×C₉₇

—

D₄×C₉₇

—

D₄×C₉₇

—

C₂×C₄×C₉₇

—

C₂×C₄×C₉₇

C₂²⋊C₄

yes

—

C₂×C₄×C₉₇

C₂²×C₄

yes

characteristic

C₂×C₄×C₉₇

C₂×D₄

yes

—

C₂×C₄×C₉₇

C₂×D₄

yes

characteristic

C₂×C₄×C₉₇

#41

yes

—

C₂×Dic₉₇

—

C₂×Dic₉₇

—

#11715895

—

C₂²×C₉₇

—

C₂²×C₉₇

—

C₂²×C₉₇

—

C₂²×C₉₇

—

C₂²×C₉₇

C₂×C₂²⋊C₄

yes

C₂²×C₉₇

yes

C₂²×C₉₇

yes

C₂²×C₉₇

yes

—

C₄×C₉₇

—

C₄×C₉₇

C₄×D₄

yes

—

C₄×C₉₇

#121

yes

—

Dic₉₇

—

C₂×C₉₇

—

C₂×C₉₇

#390

yes

characteristic

C₂×C₉₇

#418

yes

—

C₂×C₉₇

#490

yes

—

C₂×C₉₇

#496

yes

—

#1961

—

maximal, Sylow

C₉₇

#1961

yes

characteristic, complemented, Sylow

C₂×#127

D₉₇

yes

characteristic

C₂×#102

—

C₂×#97

—

#382

—

C₂²×D₄

—

C₂²×D₄

C₂×D₉₇

yes

—

C₂²×C₈

—

#127

—

#127

Dic₉₇

yes

complemented

C₂×C₄⋊C₄

—

C₂×C₄²

C₂×D₉₇

yes

characteristic

#102

—

194

—

#97

—

194

—

#95

—

194

—

C₂⁴

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

C₂×Dic₉₇

yes

—

C₂²×C₄

—

194

—

C₂²×C₄

C₂²⋊D₉₇

yes

—

C₂²×C₄

C₂²×D₉₇

yes

characteristic

C₂×C₈

—

C₂×C₈

—

194

—

C₂×C₈

—

194

—

C₄⋊C₄

—

194

—

C₄²

C₂×Dic₉₇

yes

—

C₄²

C₂²⋊D₉₇

yes

—

C₂³

—

C₂³

—

C₂³

—

100

C₂³

—

101

C₂³

C₂×C₂²⋊D₉₇

yes

characteristic

102

D₄

—

103

D₄

—

104

D₄

—

105

C₂×C₄

—

106

C₂×C₄

—

388

—

107

C₂×C₄

—

388

—

108

C₂×C₄

C₂²⋊Dic₉₇

yes

—

109

C₂×C₄

C₂²×Dic₉₇

yes

characteristic

110

C₂×C₄

D₄×D₉₇

yes

—

111

C₂×C₄

C₂×C₂²⋊D₉₇

yes

characteristic

112

C₂×C₄

#13095510

yes

—

113

C₈

—

194

—

114

C₂²

—

115

C₂²

—

116

C₂²

—

117

C₂²

—

118

C₂²

C₂×C₂²⋊Dic₉₇

yes

characteristic

119

C₂²

#15690797

yes

characteristic

120

C₂²

#15690806

yes

characteristic

121

C₂²

#15690836

yes

—

122

C₄

—

123

C₄

—

776

—

124

C₄

D₄×Dic₉₇

yes

—

125

C₄

#15690781

yes

—

126

C₂

—

127

C₂

#20260638

yes

characteristic

128

C₂

#20260740

yes

—

129

C₂

#20261099

yes

—

130

C₂

#20261122

yes

—

	subgroup	G/H
center	C₂³	C₂×C₂²⋊D₉₇
commutator	C₂×C₄×C₉₇	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₉₇
Fitting, prime radical	C₂×#127×C₉₇	C₂
socle	C₂³×C₉₇	C₂×D₄
cosocle	C₂²×C₄×C₉₇	C₂³
hypercenter	C₂×#127	D₉₇
hypocenter	C₉₇	#1961
LCS	G ▷ C₂×C₄×C₉₇ ▷ C₂×C₉₇ ▷ C₉₇
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#127

subgroup

G/H

center

C₂³

C₂×C₂²⋊D₉₇

commutator

C₂×C₄×C₉₇

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₉₇

Fitting, prime radical

C₂×#127×C₉₇

C₂

socle

C₂³×C₉₇

C₂×D₄

cosocle

C₂²×C₄×C₉₇

C₂³

D₉₇

C₉₇

G ▷ C₂×C₄×C₉₇ ▷ C₂×C₉₇ ▷ C₉₇

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#127

N	H ≅ G/N	type
1. C₂×#15690731	C₂	semidirect
2. C₂×#15690721	C₂	semidirect
3. #20260601	C₂	semidirect
13. #127×C₉₇	C₄	semidirect
18. C₂×#11715895	D₄	semidirect
66. C₉₇	#1961	semidirect
75. #127	Dic₉₇	semidirect

H ≅ G/N

type

1. C₂×#15690731

C₂

semidirect

2. C₂×#15690721

C₂

semidirect

3. #20260601

C₂

semidirect

13. #127×C₉₇

C₄

semidirect

18. C₂×#11715895

D₄

semidirect

66. C₉₇

#1961

semidirect

75. #127

Dic₉₇

semidirect

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂×#15690731	2
C₂×#15690721	2
#20260601	2
C₂×#127×C₉₇	2
#1961	97

subgroup

index

C₂×#127×C₉₇

Subgroups — #27142561

Conjugacy classes of subgroups

Complements

Maximal subgroups